A second eigenvalue bound for the Dirichlet Laplacian in hyperbolic space

2005/11/11 by Benguria, Rafael D., Linde, Helmut · 1 citation
#35P15 #49Rxx #58Jxx #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.math-ph/0511045

Abstract

Let Ω be some domain in the hyperbolic space \Hn (with n≥ 2) and S₁ the geodesic ball that has the same first Dirichlet eigenvalue as Ω. We prove the Payne-Pólya-Weinberger conjecture for \Hn, i.e., that the second Dirichlet eigenvalue on Ω is smaller or equal than the second Dirichlet eigenvalue on S₁. We also prove that the ratio of the first two eigenvalues on geodesic balls is a decreasing function of the radius.

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